Inspired from the above discussion, in this work, we answer the open question of Hölder regularity results for weak solutions to fractional \((p,q)\) -problems, for the case \(1<q\le p<2\) . We prove the interior regularity result for local weak solutions, as in theorem REF , by establishing a weak Harnack type inequality (see proposition REF ) and then we complete the proof of Hölder continuity in the spirit of [1]}. As noted earlier, for the homogeneous fractional \(p\) -Laplacian case with \(p<2\) , to prove the weak Harnack inequality, authors in [2]}, used the solution of \((-\Delta )_p^s v=1\)  in \(B_1\)  and its appropriately scaled version to construct a suitable subsolution. Nevertheless due to the lack of the scaling property, we can not rely on this method for our case. Moreover, we observe that by establishing the logarithmic lemma (in accordance with [3]} and lemma REF , below), we can not prove the Hölder continuity result by applying De Giorgi's iteration technique as in the proof of [3]}. This is due to the presence of non-homogeneous power of the parameter \(\lambda \)  (taken in the place of \(d\) , there), which prevents to choose the suitable constant \(k\)  involved in the proof of [3]}. So, we follow the idea of [6]} to prove our weak Harnack inequality. However, because of the non-homogeneous nature of the operator, we obtain two nonlocal tails with different behavior, which require special technical care at several places. We overcome the difficulty raised by these nonlocal tails by further improving the parameters involved in the proofs (by adding an additional quantity of the form \(r^{s_1p-s_2q}\) ). Our interior regularity result complements that of [3]} to the case of non-homogeneous fractional \((p,q)\) -problems with non-homogeneous right hand side in \(L^\gamma _{\rm loc}(\Omega )\) , whereas in [8]} in addition to \(q\ge 2\) , \(f\in L^\infty _{\rm loc}(\Omega )\)  is assumed. We expect that this approach, while currently focussed on a specific form of the operator, can be further applied for a wider class of nonlocal and non-homogeneous operators (including the nonlocal double phase operator) which will be the object of a separate paper by the authors. It is also worth noting that we do not assume any boundedness condition on the solution in our interior regularity result, thanks to proposition REF , which complements that of [8]}. The proof of proposition REF  relies on a Caccioppoli type inequality for non-homogeneous operators (see [8]}) and De Giorgi type iteration arguments. The main obstacle in the proof is due to \(f\in L^\gamma _{\rm loc}(\Omega )\) , for \(\gamma <\infty \) , as we can not use the Moser type iteration argument, available for the homogeneous case in [11]}. In this paper, clever convexity arguments proved that \(u_+\)  is a subsolution with right hand side \(|f|\) , whenever \(u\)  is a solution to (REF ) (with \(p=q\)  and \(s_1=s_2\) ), but seem to be inefficient in presence of non-homogeneous operators. Subsequently, using the boundary behavior of [8]}, we obtain the global Hölder continuity result for weak solutions. Additionally, employing the approach of [6]} to our case, we obtain the Harnack inequality, as in theorem REF . Similar to the homogeneous case, our Harnack inequality is valid for sign changing solution also and the sign changing behavior is displayed in terms of the nonlocal tails.
